Q.If $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is such that $A^2 = I$, then (A) $1 + \alpha^2 + \beta\gamma = 0$ (B) $1 - \alpha^2 + \beta\gamma = 0$ (C) $1 - \alpha^2 - \beta\gamma = 0$ (D) $1 + \alpha^2 - \beta\gamma = 0$

Concept: Square Matrix Definition — we compute $A^2$ directly and equate it to the identity matrix $I$.

Step 1: Compute $A^2$:

$A^2 = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} = \begin{bmatrix} \alpha^2 + \beta\gamma & \alpha\beta - \alpha\beta \\ \alpha\gamma - \alpha\gamma & \beta\gamma + \alpha^2 \end{bmatrix} = \begin{bmatrix} \alpha^2 + \beta\gamma & 0 \\ 0 & \alpha^2 + \beta\gamma \end{bmatrix}$ …

We are given a $2 \times 2$ matrix $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ with the property $A^2 = I$, where $I$ is the identity matrix. The question asks which of the four given equations must hold.

The key idea is straightforward: square the matrix, set the result equal to $I$, and compare entries. This will produce conditions on $\alpha, \beta, \gamma$. Let's do it step by step.

1. Compute $A^2$. For a $2 \times 2$ matrix, multiplication is direct:

The entry in row 1, column 1: $\alpha \cdot \alpha + \beta \cdot \gamma = \alpha^2 + \beta\gamma$.

Row 1, column 2: $\alpha \cdot \beta + \beta \cdot (-\alpha) = \alpha\beta - \alpha\beta = 0$.

Row 2, column 1: $\gamma \cdot \alpha + (-\alpha) \cdot \gamma = \alpha\gamma - \alpha\gamma = 0$.

Row 2, column 2: $\gamma \cdot \beta + (-\alpha) \cdot (-\alpha) = \beta\gamma + \alpha^2$.

So

Notice the off-diagonal entries are zero automatically — that's a nice simplification.

2. Set $A^2 = I$.

   The identity matrix is $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Equating entry by entry gives:

   • Top-left: $\alpha^2 + \beta\gamma = 1$.
   • Bottom-right: $\alpha^2 + \beta\gamma = 1$ (same condition).
   • Off-diagonals: $0 = 0$ (already satisfied).

   So the only condition is

3. Rearrange to match the options. …

Method: Deriving a condition on entries from a matrix identity like $A^2 = I$

Use this whenever a matrix with symbolic entries satisfies a stated identity ($A^2 = I$, $A^2 = A$, etc.) and you must pick the correct relation among options.

Steps

Step 1: Compute the required power/product symbolically.

Multiply the matrix out (row-by-column) keeping the symbols, so each entry of $A^2$ is an expression in the unknowns.

Step 2: Equate to the target matrix entry-by-entry. …

Common Mistakes

Mistake 1: Mishandling the sign when squaring the $-\alpha$ entry.

Why it's wrong: $(-\alpha)^2 = +\alpha^2$, so the bottom-right entry of $A^2$ is $\alpha^2 + \beta\gamma$, not $-\alpha^2 + \beta\gamma$. Correct approach: square carefully — the minus sign disappears.

Mistake 2: Sign errors when rearranging into the option form. …